Extension Operators for Sobolev Spaces on Periodic Domains, Their Applications, and Homogenization of a Phase Field Model for Phase Transitions in Porous Media von Martin H¨opker Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften { Dr. rer. nat. { Vorgelegt im Fachbereich 3 (Mathematik & Informatik) der Universit¨atBremen im Mai 2016 Datum des Promotionskolloquiums: 12. Juli 2016 Betreuer: Prof. Dr. Michael B¨ohm (Universit¨atBremen) Gutachter: Prof. Dr. Alfred Schmidt (Universit¨atBremen) Prof. Dr. Ralph E. Showalter (Oregon State University) iii Abstract The first part of this thesis is concerned with extension operators for Sobolev spaceson periodic domains and their applications. When homogenizing nonlinear partial differ- ential equations in periodic domains by two-scale convergence, the need for uniformly bounded families of extension operators often arises. In this thesis, new extension op- erators that allow for estimates in the whole domain, even if the complement of the periodic domain is connected, are constructed. These extension operators exist if the domain is generalized rectangular. They are useful for homogenization problems with flux boundary conditions. Additionally, the existence of extension operators that respect zero and nonnegative traces on the exterior boundary is shown. These can be applied to problems with mixed or Dirichlet boundary conditions. Making use of this type of extension operators, uniform Poincar´eand Korn inequalities for functions with mixed boundary values in periodic domains are proven. Furthermore, a generalization of a compactness theorem of Meirmanov and Zimin, which is similar to the well-known Lions-Aubin compactness theorem and which is applicable in periodic domains, is presented. The above results are then applied to the homogenization by two-scale convergence of some quasilinear partial differential equations and variational inequalities with operators of Leray-Lions type in periodic domains with mixed boundary conditions. In the second part, a phase field model for phase transitions on the pore scale ofa porous medium is introduced. The existence and uniqueness of weak solutions is shown. Uniform a-priori estimates are established by making use of extension operators of the type of those that have previously been constructed in the first part of this thesis. By using two-scale convergence and again applying the extension operators, a homogenized phase field model is obtained. Finally, by applying the method of formal asymptotic expansion, a macroscopic sharp interface model for phase transitions in porous media is derived. Possible applications include the melting of permafrost soil and the frost attack on concrete. iv Zusammenfassung Der erste Teil dieser Arbeit behandelt Fortsetzungsoperatoren fur¨ Sobolev-R¨aume auf periodischen Gebieten und ihre Anwendungen. Im Zusammenhang mit der Homoge- nisierung nichtlinearer partieller Differentialgleichungen in periodischen Gebieten mit Hilfe von Zwei-Skalen-Konvergenz, werden h¨aufig gleichm¨aßig beschr¨ankte Familien von Fortsetzungsoperatoren ben¨otigt. In dieser Arbeit werden neue Fortsetzungsoperatoren konstruiert, welche gleichm¨aßige Absch¨atzungen im ganzen Gebiet erlauben, selbst wenn das Komplement des periodischen Gebietes eine zusammenh¨angende Menge bildet. Diese Fortsetzungsoperatoren existieren fur¨ verallgemeinert rechteckige Gebiete und finden An- wendungen bei Homogenisierungsproblemen mit Flussrandbedingungen. Zudem werden Fortsetzungsoperatoren, die verschwindende- und nichtnegative Spuren auf dem ¨außeren Rand berucksichtigen,¨ konstruiert. Diese lassen sich auf Probleme mit gemischten- oder Dirichlet-Randbedingungen anwenden. Unter Verwendung dieser Fortsetzungsoperatoren werden gleichm¨aßige Poincar´e-und Korn-Ungleichungen fur¨ Funktionen in periodischen Gebieten mit gemischten Randbe- dingungen bewiesen. Weiterhin wird eine Verallgemeinerung eines Kompaktheitssatzes von Meirmanov und Zimin gezeigt. Dieser ist ¨ahnlich zum bekannten Lemma von Lions- Aubin und l¨asst sich im Fall periodischer Gebiete anwenden. Die obigen Resultate werden angewendet, um einige quasilineare partielle Differen- tialgleichungen und Variationsungleichungen mit Operatoren vom Leray-Lions-Typ zu homogenisieren. Dies geschieht im Kontext periodischer Gebiete und gemischter Rand- bedingungen. Im zweiten Teil wird ein Phasenfeldmodell fur¨ Phasenuberg¨ ¨ange auf der Porenebene eines por¨osen Mediums vorgestellt. Die Existenz und Eindeutigkeit schwacher L¨osungen wird gezeigt. Gleichm¨aßige a-priori Absch¨atzungen werden durch Anwendung von Fort- setzungsoperatoren gewonnen. Diese Operatoren sind dabei vom gleichen Typ wie jene, die im ersten Teil der Arbeit konstruiert wurden. Mit Hilfe von Zwei-Skalen-Konvergenz und unter Verwendung der Fortsetzungsoperatoren ergibt sich ein homogenisiertes Pha- senfeldmodell. Schließlich, durch formale asymptotische Entwicklung, wird ein makrosko- pisches Modell fur¨ Phasenuberg¨ ¨ange in por¨osen Medien hergeleitet, bei dem die Phasen durch eine singul¨are Fl¨ache getrennt werden. Zu den m¨oglichen Anwendungen der Mo- delle z¨ahlen das Auftauen von Permafrostb¨oden sowie der Frostangriff auf Beton. v Acknowledgements First of all, I want to thank my advisor Prof. Dr. Michael B¨ohmfor his great support over the last several years. His constant advice and encouragement to pursue my own ideas have made this thesis possible. I am very sorry that he has lately fallen ill, and I wish him the very best for a quick recovery. In this respect, my sincere thanks also go to Prof. Dr. Alfred Schmidt, who, on a very short notice, agreed to examine my thesis. I also thank Prof. Dr. Ralph E. Showalter for taking interest in my work and for kindly agreeing to be the second examiner of my dissertation. Furthermore, I would like to thank my colleagues from the Research Group Modeling and Partial Differential Equations, Michael Eden, Simon Gr¨utzner,and PD Dr. Michael Wolff, for all of our innumerable and valuable discussions. Especially, I thank Michael Eden and Simon Gr¨utznerfor very carefully reading the manuscript of this thesis and for providing a lot of greatly helpful remarks and suggestions. Last but not least, I thank Julitta von Deetzen for taking care of all the administrative related matters. Contents List of Figures ix List of Symbols xi 1. Introduction 1 2. Mathematical Preliminaries 5 2.1. Notation and Definitions . 5 2.2. Homogenization and Two-Scale Convergence . 7 2.2.1. Homogenization . 7 2.2.2. Two-Scale Convergence . 8 2.2.3. Stationary Two-Scale Convergence . 8 2.2.4. Time-Dependent Two-Scale Convergence . 9 2.2.5. Time-Dependent Two-Scale Convergence in Periodic Domains . 11 3. Extension Operators in Periodic Domains 13 3.1. Introduction . 13 3.2. Assumptions and Definitions . 17 3.2.1. Basic Assumptions and Definitions . 17 3.2.2. The Boundary of P E ........................ 18 \ 3.3. Statement of the Results . 21 3.4. Proofs of the Results . 22 3.4.1. Local Extension Operators . 23 3.4.2. Construction of the Global Extension Operators . 28 4. On Poincar´e'sand Korn's Inequalities for Periodic Domains and Mixed Bound- ary Conditions 35 4.1. Introduction . 35 4.2. Poincar´e'sInequality . 36 4.3. Korn's Inequality . 39 5. On the Compactness Result of Meirmanov and Zimin 45 5.1. Introduction . 45 5.2. The Compactness Result . 47 viii Contents 6. Applications 53 6.1. Introduction . 53 6.2. Preliminaries . 54 6.3. Homogenization of Elliptic Problems with Dirichlet or Mixed Boundary Conditions in Periodic Domains . 55 6.3.1. The Elliptic Operators and Their Properties . 55 6.3.2. Elliptic Equations . 59 6.3.3. Elliptic Variational Inequalities . 61 6.4. Homogenization of Parabolic Equations with Dirichlet or Mixed Boundary Conditions in Periodic Domains . 66 6.4.1. Mathematical Preliminaries . 66 6.4.2. Homogenization . 68 7. Homogenization of a Phase Field Model for Phase Transitions in Porous Media 71 7.1. Introduction . 71 7.2. Modeling Phase Transitions . 73 7.3. The Microscopic Problem . 76 7.3.1. Definitions and Assumptions . 76 7.3.2. Classical Formulation . 79 7.3.3. Weak Formulation . 80 7.4. Existence . 82 7.4.1. Operator Formulation . 82 7.4.2. Mathematical Preliminaries . 83 7.4.3. Existence Proof . 86 7.5. Uniqueness . 104 7.6. Homogenization . 108 7.6.1. Preliminaries . 108 7.6.2. The Exterior Boundary . 108 7.6.3. Passing to the Limit . 110 7.6.4. Derivation of Homogenized Diffusion Tensors . 113 7.6.5. Properties of the Homogenized Diffusion Tensors . 115 7.6.6. The Full Homogenized Model . 116 7.6.7. Uniqueness . 117 7.7. Sharp Interface Limit of the Homogenized Model . 120 7.7.1. Scaling . 120 7.7.2. Parametrization of the Interface . 121 7.7.3. Outer Expansions . 122 7.7.4. Inner Expansions . 123 7.7.5. The Sharp Interface Model . 127 8. Summary and Outlook 129 Appendix A. Density of Smooth Functions in Sobolev Spaces 131 Appendix B. Monotone Operators 135 Bibliography 137 List of Figures 2.1. A 1-periodic medium E and the "-periodic domain Ω = Ω ("E). 6 " \ 2.2. The reference cell Y and its part Z = Y E. ................ 7 \ 2.3. The exterior boundary of a periodic domain. 7 3.1. Example of a reference cell in the pipe model of porous media. 14 3.2. Example of a generalized rectangular domain Ω. 15 3.3. Examples of AO's and their respective translations into "−1Ω. 18 3.4. Classification of the sets Ui........................... 19 3.5. A set Ui of Type 2. 19 3.6. The mapping Φi, corresponding to a set Ui of Type 2. 20 4.1. Construction of the auxiliary domain Ω.e . 43 7.1. The geometry of the Stefan problem. 73 7.2. The transition of the phase field variable 휒 across the phase interface, and the double-well potential g. .......................... 75 7.3. Schematic of the reference cell. 76 s 7.4. Schematic of the porous Medium Ω, consisting of the solid matrix Ω" and p the pore space Ω" . ............................... 77 List of Symbols (Strong) convergence, p. 6 ! * Weak convergence, p. 9 * * Weak-* convergence, p. 97 2 Two-scale convergence, p. 8 ! n ei The i-th unit vector in R , p. 6 xT Transpose of a vector or matrix, p.